Question

If \[ (4^{\sec^{2}\alpha})x^{2}+2x+\left(\beta^{2}-\beta+\frac{1}{2}\right)=0 \] has real roots, then the value/value(s) of \[ (\cos\alpha+\cos^{-1}\beta) \] is/are:

• A. $1+\frac{\pi}{3}$ if $\alpha = 2n\pi$
• B. $-1-\frac{\pi}{3}$ if $\alpha = (2n+1)\pi$
• C. $-1+\frac{\pi}{3}$ if $\alpha = (2n+1)\pi$
• D. $-1+\frac{\pi}{3}$ if $\alpha = 2n\pi$

Hint:

Whenever a discriminant inequality forces a condition like $\text{Expression} \le 1$, and your component factors have minimum values that multiply to exactly 1, the inequalities lock up completely! The system collapses from an infinite range down to a single fixed intersection point.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A quadratic equation \( Ax^2 + Bx + C = 0 \) has real roots if the discriminant \( D = B^2 - 4AC \ge 0 \). We use this inequality alongside the ranges of trigonometric and quadratic expressions to find unique values for the variables.
Step 2: Key Formula or Approach:
1. \( 2^2 - 4(4^{\sec^2 \alpha})(\beta^2 - \beta + 1/2) \ge 0 \).
2. Solve the resulting inequality \( 4^{\sec^2 \alpha}(\beta^2 - \beta + 1/2) \le 1 \).
Step 3: Detailed Explanation:
Consider the term \( A = 4^{\sec^2 \alpha} \). Since \( \sec^2 \alpha \ge 1 \), \( A \ge 4^1 = 4 \).
Consider the term \( C = \beta^2 - \beta + 1/2 = (\beta - 1/2)^2 + 1/4 \). The minimum value of C is \( 1/4 \).
The discriminant condition \( AC \le 1 \).
But from our analysis: \( \text{Min}(A \cdot C) = 4 \times (1/4) = 1 \).
The only way \( AC \le 1 \) can hold is if \( AC = 1 \) exactly, meaning both terms must be at their minimum.
1. \( 4^{\sec^2 \alpha} = 4 \implies \sec^2 \alpha = 1 \implies \alpha = n\pi \).
2. \( (\beta - 1/2)^2 + 1/4 = 1/4 \implies \beta = 1/2 \).
Now calculate the value \( \cos \alpha + \cos^{-1} \beta \):
\( \cos^{-1}(1/2) = \pi/3 \).
If \( \alpha = 2n\pi \) (even multiple), \( \cos \alpha = 1 \). Sum \( = 1 + \pi/3 \). (A) is correct.
If \( \alpha = (2n+1)\pi \) (odd multiple), \( \cos \alpha = -1 \). Sum \( = -1 + \pi/3 \). (C) is correct.
Step 4: Final Answer:
The possible values are \( 1 + \pi/3 \) and \( -1 + \pi/3 \).